question_answer

A ray of light is incident on the hypotenuse of a right-angled prism after travelling parallel to the base inside the prism. If \[\mu \] is the refractive index of the material of the prism, the maximum value of the base angle for which light is totally reflected from the hypotenuse is

A) \[{{\sin }^{-1}}\,\left( \frac{1}{\mu } \right)\]                     

B) \[{{\tan }^{-1}}\,\left( \frac{1}{\mu } \right)\]

C) \[{{\tan }^{-1}}\,\left( \frac{\mu -1}{\mu } \right)\]

D) \[{{\cos }^{-1}}\,\left( \frac{1}{\mu } \right)\]

Correct Answer: D

Solution :

If \[\alpha \] = maximum value of base angle for which light is totally reflected form hypotenuse.   \[\left( 90{}^\circ  - \alpha  \right) = C =\] minimum value of angle of incidence at hypotenuse for total internal reflection \[\sin  \left( 90{}^\circ  - \alpha  \right) = sin C =\frac{1}{\mu }\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\cos \alpha =\frac{1}{\mu }\] \[\Rightarrow \,\,\,\,\,\,\alpha ={{\cos }^{-1}}\left( \frac{1}{\mu } \right)\,\]